Questions tagged [topology]

In mathematics, topology examines the properties of space (such as connectedness and compactness) that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing.

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Lie group and corresponding smooth manifold, and also why $SO(3)$ have a 3-dimensional manifold embedded in 4-dimensional Real space?

I think I have some loop holes on a connecting smooth manifold to a lie group. I state what my concepts are, Lie groups are expressed as manifold because the parameters in corresponding metric form a ...
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Is a Kerr black hole related to the Hopf fibration?

The Hopf fibration is a way of divided space into circles. There is a planar circle at the "centre" with other circle wrapped round it in a stack of toroidal surfaces. The Kerr rotating black hole it ...
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What is the meaning of a flat metric invariant up to a Weyl factor?

When studying CFT I was told that: A conformal transformation is a spacetime transformation that leaves the flat metric invariant up to a Weyl factor. What is the meaning of leaving a metric ...
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Are the spacelike foliations of a non-static spacetime topologically equivalent?

Assuming a stationary, globally hyperbolic spacetime, I can imagine that all spacelike foliations are topologically equivalent though not all will be identical since the spacetime is not static. Is ...
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Is a reasonable spacetime geodesically connected?

There are some theorems concerning whether a spacetime is geodesically connected (whether any two points $p, q \in M$ admit a geodesic connecting them) or not, ie [1][2], but all of these are ...
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Consistency condition for Yang-Mills on a Torus

So I was recently studying 't Hooft's paper on self-dual solutions to Yang-Mills on $\mathbb{T}^4$. So the basic idea is that you consider a box with periodic boundary conditions and then you impose ...
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1answer
100 views

Why are magnetic monopoles hard to find (if exist)?

I understand the Yang-Mill perspective of $U(1)$-gauge theory. In that, you can easily write down the field of a Dirac magnetic monopole. What interests me is the fact that it's so hard to find (if ...
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What are the physics behind the hairy ball theorem?

What are the physics behind the hairy ball theorem? It seems that this theorem is a corollary of the way that vector fields are constructed or what happens to numbers when we move from three to two ...
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Is it possible to model spacetime as a function from an interval $I$ to a set $S$, where $t\in I$ represents time and $x\in S$ represents space? [duplicate]

According to wikipedia, spacetime is modeled via a pseudo-Riemannian 4-manifold, with one dimension representing time and the other three dimensions representing space. I would like to know whether ...
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54 views

How do the Euler-Lagrange equations generalise to an arbitrary manifold?

So every formalism for the EL equations I have seen relies on choosing a coordinate chart. However, if we had say, a field on a sphere, then we can’t have global coordinates. How, in principle, ...
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1answer
59 views

Momentum shift after threading magnetic flux through a ring

The question in short: A translationally-invariant system living on a ring is in a state of momentum $p_0$. How does the momentum change after threading one magnetic flux quantum through the ring?...
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1answer
65 views

Alternatives for calculating topological invariants in topological materials

My questing is regarding the different alternatives for calculating topological invariants in topological materials protected by symmetry, specially time-reversal invariant topological insulators, ...
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What kind space does spinor lives in?

I'm trying to read some differential geometry these days and I just encountered orientable manifold. Quote: "If $M$ is nonorientable, $M$ has a two-sheeted orientable covering manifold $\tilde{M}$. ...
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130 views

Does visible universe have shape of a 3-sphere?

Here's my logic: If you look out in the visible universe you see further back in time. Look enough back and you get to the big bang singularity. This means whichever way you look in the visible ...
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1answer
70 views

Theta-terms in 3+1D QCD and 1+1D QED / $CP^1$ models

It is well known that topological $\theta$-terms in gauge theories are total derivatives and vanish after integration over the Lagrangian (or Hamiltonian) density, unless there are nontrivial boundary ...
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What are the topological properties of a Schwarzschild black hole, and of its horizon and singularity?

What is the topology of a black hole spacetime? What about its horizon and its singularity?
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Number of Weyl points due to symmetries

In Weyl semimetals time reversal symmetry (TRS) or centrosymmetry has to be broken (CS). It is stated, there are at least four Weyl points in TRS systems. I tried understanding this looking at the ...
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1answer
47 views

Local density in SSH model

Considering the usual SSH model defined on N sites and parametrized with $\delta $: $$H = \sum\limits_{j=0}^{N-1} (1-\delta)\ c_{j,A}^{\dagger}\ c_{j,B} +(1+\delta)\ c_{j,B}^{\dagger}\ c_{j+1,A} + h.c ...
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Can the interior of a black hole be treated as a non-metrizable or non-Hausdorff space?

Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated. https://en.wikipedia.org/wiki/Subspace_topology My question is regarding how ...
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2answers
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Physical Systems whose Phase Space is not a Cotangent Bundle

I'm trying to justify the full power of symplectic mechanics yet I keep finding examples of physical systems which are only trivial examples of symplectic mnaifolds, cotangent bundles. What physical ...
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On earth, is a levelled ground perpendicular to the radius of earth?

During construction of a building we level the ground using laser or more simply a water pipe. Method using water pipe: We level the ground such that the water level is same at all four corners ...
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Orientability of compactified manifolds in string theory

Calabi-Yau manifolds in string theory are orientable (topologically you can make "handed" structures in them). Non-orientable manifolds are perfectly respectable (the projective plane is a basic ...
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68 views

$SU(2)$ vs $SO(3)$ in Quantum Mechancs

When we're talking about spatial rotations is quantum mechanics, why do we need to resort to $SU(2)$? Why isn't $SO(3)$ enough? I've read that $SO(3)$ isn't simply connected, and I've read about ...
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Why are covering groups more 'fundamental'?

So I understand that the Lie algebra of $SO(1,3)$ is isomorphic to the Lie algebra of $SU(2)\oplus SU(2)$, and the Lie algebra of $SO(3)$ is isomorphic to one copy of $SU(2)$ (at the group level we ...
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3+1 foliations of a spacetime [duplicate]

I would like to know can any Lorentzian manifold be 3+1 foliated? I would like to know when a 3+1 foliation of a Lorentzian manifold exists and when it does not exist. If a 3+1 foliation does not ...
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Polyakov action with Kalb-Ramond field defined globally?

In string theory, with the addition of the anti-symmetric $B$-field, the Polyakov action takes the form: $$S=\frac{1}{4 \pi \alpha^{'}} \int_{\sum}d \sigma d \tau (\cdots + \epsilon^{\alpha \beta} B_{\...
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QCD generating functional and QCD vacuum from nonperturbative to perturbative regime!

The complete generating functional in QCD (starting from the most general renormalizable, Lorentz invariant and gauge invariant Lagrangian) given by $$Z_\theta[J]=\int \mathcal{D}A \exp i\int d^4x~ {\...
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107 views

Where do theta terms live?

Consider a gauge theory with group $G$. The canonical kinetic term for the gauge field is $F\wedge\star F$ and, depending on the dimensionality of spacetime, there are other allowed terms, such as ...
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1answer
29 views

Regarding the skin effect, is the inner surface of a tube a skin?

According to the skin effect, alternating current in a conductor has the highest current density on the surface, and it drops with the distance from the surface, such that the current is conducted in ...
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Can we tell the difference between a scalar field and a non-linear sigma model?

Suppose a $U(1)$ non-linear sigma model field $\Sigma(x)$ take values on a circle. But if this circle is very large and the value don't vary so much, shouldn't this be almost identical to a scalar ...
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1answer
57 views

Polchinski's String theory Green's function on $RP_2$; eq. (6.2.38) p. 176

Is there an error in Polchinski's String Theory Equation (6.2.38) p 176 that is not on the Errata's page https://www.kitp.ucsb.edu/joep/links/joes-big-book-string/errata ? He write for the Green's ...
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68 views

Determining geometry/topology from a Line Element

Is it possible given a line element, to determine its geometry? For example whether the line element $ds^2$ of a 2D surface corresponds to $\mathbb{R}^2$ or $S^2$ geometry?
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Existence of spin-$\frac{1}{2}$ representation corresponds to $\text{SO}(3)$ having double cover?

I come across this article: https://skullsinthestars.com/2016/03/29/1975-neutrons-go-right-round-baby-right-round/ I quote here a part of this article: Spin 1/2 particles like the electron, ...
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1answer
30 views

I am trying to integrate the Berry curvature of Dirac Semimetal to obtain anomalous hall conductivity

$$\Omega_{ij}(k)^{\pm} = \frac{v^2cos(\theta)\left(b\pm\frac{vk_z^2}{\sqrt{k_y^2\sin^2(\theta) +k_z^2}}\right)}{4\left(v^2k_x^2+v^2k_y^2cos^2(\theta)+\left( b\pm v\sqrt{k_y^2\sin^2(\theta) +k_z^2}\...
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Berry curvature of a two band lattice Hamiltonian

Consider a general two band lattice Hamiltonian, which can be expressed as a $2\times2$ matrix: $$\mathcal{H}=\vec{h}(\vec{k})\cdot \vec{\sigma}$$ where $\vec{\sigma}$ is the vector of Pauli matrices. ...
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1answer
78 views

Uses of Homotopy theory in Condensed Matter Physics [closed]

I have started to read the homotopy theory, basically from Nakahara's book. I am mainly interested in condensed matter physics. But, I don't know at the end of the day what are the problems I'll be ...
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1answer
91 views

Why do we need conformal compactification to define the global conformal group?

First I have the definition of a conformal map. Let $(M,g)$ and $(M',g')$ be two pseudo-Riemannian manifolds of same dimension. Let $U\subset M$ and $V\subset M'$, we say that a smooth map of maximal ...
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143 views

Feynman diagrams as topology

When we talk about Feynman diagrams we know they are tools to make calculations easier and more intuitive. Moreover, it's said that they are "topological" representations of the interactions. But, ...
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Superfluid Vortex Lines Bifurcation and Winding Numbers

Vortex lines in superfluids are characterized by their quantised circulation: $k = \frac{h}{m}\times n$, where $n$ is the winding number in the sense of a topological winding number. Now, most vortex ...
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1answer
69 views

Positive local spatial curvature of the universe implies that the universe is compact (i.e. finite)?

I quote from the Wikipedia page about the shape of the universe: If the spatial geometry [of the universe] is spherical, i.e., possess positive curvature, the topology is compact. I'm trying to ...
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2answers
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Are there winding-number vacua in Weinberg-Salam (Or are they a gauge artifact)?

In pure SU(2) Yang-Mills the vacua van be grouped in homotopy classes labeled by their winding number. Instantons connect these giving rise to the theta-vacuum. I’m studying the SU(2) sphaleron in ...
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1answer
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Can a “time dimension” be part of a spherical topology?

I've heard it speculated that the spatial dimensions of the universe is a 3-sphere. Or a 3-torus. But usually, I guess, it's assumed that the "time" dimension just has its own geometry, like a line, ...
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35 views

Bounds on the size of the normal neighbourhood

One of the most important feature of general relativity is the existence of the convex normal neighbourhood, a neighbourhood $U$ on which the exponential map is a diffeomorphism between $\exp^{-1}(U)$ ...
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3answers
131 views

Notion of Present

Can't I sync all watches in spacetime and call this time slice the present? In Carlo Rovelli's book he tried to explain that the notion of the present is local only, which I could not follow.
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Is Chern number still well defined with band touching?

Consider a 2 band system in 2d with band crossing on a ring. The coupling opens a gap. If the coupling is zero at some points of the ring, the band is still touching at these points. The berry ...
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How build a topological charge from of the mapping between physical and inner space?

How build a topological charge from the mapping between physical and inner space? When we make a mapping between two coordinates system, we normally relate both systems by coordinate transformation as,...
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1answer
58 views

What does topologically stable mean?

I am working on an article about skyrmion manipulation and it is written that those particles are "topologically stable particle-like spin configurations that carry a characteristic topological charge ...
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63 views

Connected and disconnected dimensions

The usual way of determining the dimensionality of space is from the number of values needed to define a unique point. However, when choosing a ski, my body is defined by two numbers - my mass and ...
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2answers
81 views

In the most trivial spacetimes, is the existence of a null geodesic equivalent to horismos relations?

Take a globally hyperbolic topologically trivial spacetime $M \cong \mathbb{R} \times \Sigma$, $\Sigma \cong \mathbb{R}^{(n-1)}$. Given $p, q \in M$, such that there exists a future-directed null ...
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35 views

Conditions for zero mode edge state to appear

Consider a non-interacting translationally invariant system described by H(k), k is the crystal momentum. The dimensionality of the system is denoted as d. I was thinking about what are the general ...

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